Metode Eliminasi dan Substitusi dalam Menyelesaikan Sistem Persamaan Linear Tiga Variabel

The realm of mathematics often presents us with intricate systems of equations, demanding our analytical prowess to unravel their hidden solutions. Among these systems, linear equations with three variables pose a unique challenge, requiring a systematic approach to find the values that satisfy all equations simultaneously. Two powerful techniques, elimination and substitution, stand as cornerstones in tackling such systems, offering distinct yet complementary pathways to arrive at the desired solution. This article delves into the intricacies of these methods, illuminating their application in solving systems of linear equations with three variables.

Understanding the Essence of Elimination

The elimination method, as its name suggests, aims to eliminate one variable at a time from the system of equations. This is achieved by manipulating the equations through addition or subtraction, strategically combining them to eliminate a chosen variable. The process involves a series of steps, each designed to simplify the system until a single equation with a single variable remains. This final equation can then be solved directly, and the solution can be back-substituted into the previous equations to determine the values of the other variables.

The Art of Substitution

In contrast to elimination, the substitution method focuses on expressing one variable in terms of the others. This involves isolating a variable in one equation and substituting its equivalent expression into the remaining equations. This substitution effectively reduces the number of variables in the system, ultimately leading to a single equation with a single variable. Solving this equation reveals the value of one variable, which can then be substituted back into the original equations to determine the values of the remaining variables.

Illustrative Examples

To solidify our understanding of these methods, let's consider a concrete example. Suppose we have the following system of linear equations:

```

2x + y - z = 5

x - 2y + 3z = 1

3x + 4y - 2z = 8

```

Using Elimination:

1. We can eliminate *x* by multiplying the second equation by -2 and adding it to the first equation:

```

2x + y - z = 5

-2x + 4y - 6z = -2

------------------

5y - 7z = 3

```

2. Similarly, we can eliminate *x* by multiplying the second equation by -3 and adding it to the third equation:

```

3x + 4y - 2z = 8

-3x + 6y - 9z = -3

------------------

10y - 11z = 5

```

3. Now we have a system of two equations with two variables:

```

5y - 7z = 3

10y - 11z = 5

```

4. We can eliminate *y* by multiplying the first equation by -2 and adding it to the second equation:

```

-10y + 14z = -6

10y - 11z = 5

------------------

3z = -1

```

5. Solving for *z*, we get *z* = -1/3.

6. Substituting *z* = -1/3 into the equation 5y - 7z = 3, we get *y* = 2/3.

7. Finally, substituting *z* = -1/3 and *y* = 2/3 into the original equation 2x + y - z = 5, we get *x* = 4/3.

Therefore, the solution to the system of equations is *x* = 4/3, *y* = 2/3, and *z* = -1/3.

Using Substitution:

1. We can solve the second equation for *x*:

```

x - 2y + 3z = 1

x = 2y - 3z + 1

```

2. Substituting this expression for *x* into the first and third equations, we get:

```

2(2y - 3z + 1) + y - z = 5

3(2y - 3z + 1) + 4y - 2z = 8

```

3. Simplifying these equations, we get:

```

5y - 7z = 3

10y - 11z = 5

```

4. Now we have a system of two equations with two variables, which can be solved using the elimination method as described above.

Conclusion

The elimination and substitution methods provide powerful tools for solving systems of linear equations with three variables. Each method offers a distinct approach, allowing us to manipulate the equations strategically to isolate variables and ultimately arrive at the solution. While the elimination method focuses on eliminating variables through addition or subtraction, the substitution method involves expressing one variable in terms of the others. Both methods are equally valid and effective, and the choice between them often depends on the specific structure of the system of equations and personal preference. By mastering these techniques, we gain the ability to navigate the complexities of linear equations and unlock the hidden solutions they hold.